Random Sampling: Identify and Justify the Method

Random sampling is a rule for choosing observations, not a promise that every sample will look exactly like the population. Describe the rule carefully enough that another person could carry it out. Then ask whether its structure fits the question and population.

Identify the Selection Rule

Sampling without replacement means an observational unit can be selected only once. After selection, the unit is not returned to the pool for later selections, so it cannot be selected again. Sampling with replacement returns the unit to the pool before the next selection. A unit can then be selected more than once. This distinction concerns what happens after each draw, not whether every possible unit had a chance on the first draw.

A simple random sample, or SRS, of size $n$ gives every possible sample of that size the same chance of selection. Equal chances for individual units alone are not enough to identify an SRS: a method might allow only certain combinations. Label the population list, use a random-number generator to select distinct labels for a without-replacement SRS, or mix identically sized numbered slips and draw without returning them. The chance mechanism must determine the selection.

Suppose a school has 600 students in three grades. A stratified random sample first divides all students into non-overlapping groups, called strata, using a shared attribute such as grade. Students within a stratum are similar in the attribute used to form it. Select an SRS within each grade, then combine the selected students into one sample. Every stratum contributes observations.

A cluster random sample divides the population into smaller groups and selects an SRS of those groups. Collect data from all units in the selected clusters. Ideally each cluster reflects the population’s variety and clusters are similar to one another. A homeroom containing a mix of grades could be a cluster if that mix reflects the school. Choosing clusters saves trips or administrative work, but similar students within a cluster can reduce the variety of information obtained.

A systematic random sample uses a random starting point and then a fixed interval between selected units. For 600 ordered students and a sample of 30, randomly choose a starting position from 1 through 20 and take that student, then every 20th student. Without the random start, “every 20th” omits an important part of the method.

Method	First choice	Units measured
SRS	A random set of individual labels	Those selected individuals
Stratified random	Divide all units by a shared characteristic	An SRS from every stratum
Cluster random	An SRS of population groups	All units in selected groups
Systematic random	Random start	Units separated by a fixed interval

Worked Example: Strata and Clusters Are Different

A workplace has 120 people in each of three job categories. Selecting 12 people randomly within each category is stratified sampling. Selecting three work teams randomly and measuring every member of those teams is cluster sampling. Selecting 36 labels from the complete employee roster is an SRS. The same desired sample size does not make these selection rules interchangeable.

Choose a Method for the Question

Each method has characteristics that can make it useful for a particular question. If the school must compare grade-specific opinions, stratifying by grade ensures each grade appears and can give enough observations for that comparison. If interviewers must travel to scattered neighborhoods, sampling whole clusters may reduce travel. An SRS is straightforward with a complete roster and provides a clear chance mechanism. A systematic sample is convenient for a long ordered list, but the ordering deserves attention: if a repeating pattern matches the selection interval, a sample may capture an unrepresentative slice.

Worked Example: A Seat-Quality Inspection

A theater has 20 rows of 30 seats. If damage varies strongly by distance from the stage, classify rows into front, middle, and rear strata and randomly sample seats in each region. If inspectors must test every seat in a row once they set up their equipment, randomly selecting rows and testing all 30 seats in each selected row may be more efficient. The first choice favors representation across known regions; the second favors efficient group-level inspection. A justification names the tradeoff in this theater, rather than saying one method is always best.

Random sampling addresses deliberate or voluntary selection, but it does not automatically repair a missing population list, inaccurate measurements, or failure to obtain selected responses. State what the chance mechanism does and what other parts of the study still require care.

Independent Practice

Use the information in each question to make your own calculation, representation, or argument. Explain what your result means in its setting.

Original generated practice for the Fall 2026 course.

What happens after a draw?

A researcher draws an apartment number from a box. Procedure A sets the slip aside. Procedure B returns and remixes the slip before each next draw.

Part a

Identify each replacement rule and explain whether the same apartment can be selected again.

Your reasoning

Worked answer and rubric

Equal combinations

Ten students have labels 01–10. Method A uses a random-number generator to choose two distinct labels uniformly from all pairs. Method B flips a coin and selects either students 01/02 or students 09/10.

Part a

Identify which method is an SRS of size 2 and explain using possible samples. Give an equivalent numbered-slip procedure.

Your reasoning

Worked answer and rubric

Which groups are sampled?

A school divides students into grades and samples 15 students randomly from each grade. A different school randomly chooses four mixed-grade homerooms and measures every student in those homerooms.

Part a

Identify the two methods. Explain the grouping characteristic and which units are measured in each.

Your reasoning

Worked answer and rubric

The starting point matters

A list contains 500 households. An interviewer randomly selects a start from 1–10, then selects that position and every tenth position.

Part a

Identify the method and its two essential selection features. Explain a concern if every tenth list entry has a special characteristic.

Your reasoning

Worked answer and rubric

Choose a museum survey

A museum has 600 annual members: 200 family, 250 individual, and 150 student memberships. The roster has a unique label for each member. It also lists 30 neighborhood groups of 20 members each. Every neighborhood group contains a varied mix of membership categories. Each neighborhood group has approximately the same family/individual/student proportions as the overall membership. The museum wants to compare opening-time preferences by membership category and can obtain 60 responses.

Part a

Identify and describe a size-60 SRS procedure. Explain the equal-sample condition, and state whether you would sample with or without replacement.

Your reasoning

Part b

Identify a stratified random method using the three membership categories. Give one valid allocation totaling 60 and explain how the sample is assembled.

Your reasoning

Part c

Identify a cluster method that yields 60 measured members. Explain why the group descriptions are desirable for this method.

Your reasoning

Part d

Identify a systematic random method of size 60 from the ordered 600-member roster. Specify the random start and interval.

Your reasoning

Part e

Justify a choice for comparing membership categories. Explain a different circumstance in which cluster sampling could be preferable.

Your reasoning

Part f

If the museum used replacement for repeated draws of member labels, what would change after each draw and what would become possible?

Your reasoning

Worked answer and rubric